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Question

If f(x)=2x45x3+x2+3x2 is divided by g(x) the qoutient is q(x)=2x25x+3andr(x)=2x+1 find g(x).

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Solution


The division algorithm states that Dividend=Divisor×Quotient+Remainder that is f(x)=g(x)q(x)+r(x)

Here, it is given that the dividend is f(x)=2x45x3+x2+3x2, the divisor is 2x25x+3 and the remainder is 2x+1, therefore, by applying division algorithm we have:

2x45x3+x2+3x2=(2x25x+3)g(x)+(2x+1)2x45x3+x2+3x2(2x+1)=(2x25x+3)g(x)2x45x3+x2+3x2+2x1=(2x25x+3)g(x)2x45x3+x2+5x3=(2x25x+3)g(x)g(x)=2x45x3+x2+5x32x25x+3

Let us now divide 2x45x3+x2+5x3 by 2x25x+3 as shown in the above image:

From the division, we observe that the quotient is x21 and the remainder is 0.

Hence, the quotient q(x)=x21.

1237960_1096373_ans_0730b230cf564838a2dfb8623acd34f4.jpg

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